The convex hull of a given set of points, S, is the minimum number of points in S that create a convex polygon containging all of the points in S.
Why is it important: (The notion of "Variation Diminishing")
When creating curves based on control points, it is important to be able to specify certain behavioral aspects of the curve. An important aspect to quantify is whether or not the curve will remain "inside the polygon formed by the control points that describe it. As is readily apparent with the Hermite curve, this is not the case. However, with (non-rational) b-splines the curve will always be within the convex hull of the control points that affect the curve at any given time. This can readily be seen by selecting the "Toggle Convex Hull" button on the "Curve Editor" panel, then pressing the play ">>" button. Notice that the convex hull always contains the point on the curve.